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Thou shalt buy and hold


  • Albert Shiryaev
  • Zuoquan Xu
  • Xun Yu Zhou


An investor holding a stock needs to decide when to sell it over a given investment horizon. It is tempting to think that she should sell at the maximum price over the entire horizon, which is however impossible to achieve. A close yet realistic goal is to sell the stock at a time when the expected relative error between the selling price and the aforementioned maximum price is minimized. This problem is investigated for a Black-Scholes market. A stock 'goodness index' α, defined to be the ratio between the excess return rate and the squared volatility rate, is employed to measure the quality of the stock. It is shown that when the stock is good enough, or to be precise when α ≥ 1/2, the optimal strategy is to hold on to the stock, selling only at the end of the horizon. Moreover, the resulting expected relative error diminishes to zero when α goes to infinity. On the other hand, one should sell the stock immediately if α ≤ 0. These results justify the widely accepted financial wisdom that one should buy and hold a stock - if it is good, that is.

Suggested Citation

  • Albert Shiryaev & Zuoquan Xu & Xun Yu Zhou, 2008. "Thou shalt buy and hold," Quantitative Finance, Taylor & Francis Journals, vol. 8(8), pages 765-776.
  • Handle: RePEc:taf:quantf:v:8:y:2008:i:8:p:765-776 DOI: 10.1080/14697680802563732

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    References listed on IDEAS

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    Cited by:

    1. R'emy Chicheportiche & Jean-Philippe Bouchaud, 2013. "Some applications of first-passage ideas to finance," Papers 1306.3110,
    2. Xiongfei Jian & Xun Li & Fahuai Yi, 2014. "Optimal Investment with Stopping in Finite Horizon," Papers 1406.6940,
    3. Tim Leung & Xin Li & Zheng Wang, 2015. "Optimal Multiple Trading Times Under the Exponential OU Model with Transaction Costs," Papers 1504.04682,
    4. Dmitry B. Rokhlin, 2016. "Minimax perfect stopping rules for selling an asset near its ultimate maximum," Papers 1601.00175,, revised Jul 2016.
    5. Tim Leung & Xin Li & Zheng Wang, 2014. "Optimal Starting-Stopping and Switching of a CIR Process with Fixed Costs," Papers 1411.6080,
    6. repec:wsi:ijtafx:v:20:y:2017:i:03:n:s0219024917500182 is not listed on IDEAS
    7. Eddie C. M. Hui & Sheung-Chi Phillip Yam, 2014. "Can we beat the "buy-and-hold" strategy? Analysis on European and American securitized real estate indices," International Journal of Strategic Property Management, Taylor & Francis Journals, vol. 18(1), pages 28-37, March.
    8. Christoph Kuhn & Budhi Arta Surya & Bjorn Ulbricht, 2014. "Optimal Selling Time of a Stock under Capital Gains Taxes," Papers 1501.00026,
    9. repec:spr:fininn:v:3:y:2017:i:1:d:10.1186_s40854-017-0080-y is not listed on IDEAS
    10. repec:kap:jrefec:v:56:y:2018:i:1:d:10.1007_s11146-016-9564-1 is not listed on IDEAS
    11. Eddie C. M. Hui & Sheung-Chi Phillip Yam & Si-Wei Chen, 2011. "Shiryaev-Zhou index -- a noble approach to benchmarking and analysis of real estate stocks," International Journal of Strategic Property Management, Taylor & Francis Journals, vol. 16(2), pages 158-172, September.
    12. Pieter C. Allaart, 2009. "A general "bang-bang" principle for predicting the maximum of a random walk," Papers 0910.0545,


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